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Exact solutions to nonlinear nonautonomous space-fractional diffusion equations with absorption.

E K Lenzi1, G A Mendes, R S Mendes

  • 1Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá-PR, Brazil.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 6, 2003
PubMed
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This study presents new exact solutions for fractional diffusion equations with absorption, extending known models like the porous medium and thin film equations. The findings connect to maximum entropy principles using Tsallis entropy.

Area of Science:

  • Nonlinear Partial Differential Equations
  • Fractional Calculus
  • Mathematical Physics

Background:

  • Fractional diffusion equations model anomalous diffusion processes.
  • Existing models include porous medium and thin film equations.
  • Absorption terms introduce complexity in diffusion dynamics.

Purpose of the Study:

  • To derive new exact solutions for a nonlinear fractional diffusion equation with absorption.
  • To extend the applicability of fractional diffusion models.
  • To explore connections between derived solutions and maximum entropy principles.

Main Methods:

  • Analysis of a nonlinear fractional diffusion equation using fractional spatial derivatives.
  • Investigation of specific diffusion coefficients D(x,t)=D(t)/x/(-theta) and drift forces F=-k(1)(t)x+k(alpha)x/x/(alpha-1).

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  • Relating solutions to those derived from the maximum entropy principle via Tsallis entropy.

    • Main Results:

    • Obtained several new exact classes of solutions for the fractional diffusion equation.
    • Demonstrated that the employed equation generalizes known diffusion equations.
    • Established a link between the derived solutions and the Tsallis entropy-based maximum entropy principle.

    Conclusions:

    • The study successfully provides novel exact solutions for a complex fractional diffusion model.
    • The findings broaden the scope of fractional diffusion equation applications.
    • A significant connection is established between fractional calculus solutions and information theory principles.